Almost sure convergence of the Bartlett estimator |
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| Authors: | István Berkes Lajos Horváth Piotr Kokoszka Qi-man Shao |
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| Affiliation: | (1) Alfréd Rényi Institute of Mathematics, A. Rényi Institute of Mathematics Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest Hungary, Budapest;(2) Department of Mathematics University of Utah, 155 South 1440 East Salt Lake City, UT 84112-0090 USA;(3) Department of Mathematics and Statistics Utah State University, 3900 Old Main Hill Logan, UT 84322-3900 USA;(4) Department of Mathematics University of Oregon, Eugene, OR 97403-1222 USA |
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| Abstract: | Summary We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a. s. behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants and assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator. |
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| Keywords: | weak dependence long-range dependence variance of the mean cumulants increments of partial sums |
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